Numerical solution of acoustic scattering by finite perforated elastic plates

Abstract

We present a numerical method to compute the acoustic field scattered by finite perforated elastic plates. A boundary element method is developed to solve the Helmholtz equation subjected to boundary conditions related to the plate vibration. These boundary conditions are recast in terms of the vibration modes of the plate and its porosity, which enables a direct solution procedure. A parametric study is performed for a two-dimensional problem whereby a cantilevered perforated elastic plate scatters sound from a point quadrupole near the free edge. Both elasticity and porosity tend to diminish the scattered sound, in agreement with previous work considering semi-infinite plates. Finite elastic plates are shown to reduce acoustic scattering when excited at high Helmholtz numbers k₀ based on the plate length. However, at low k₀, finite elastic plates produce only modest reductions or, in cases related to structural resonance, an increase to the scattered sound level relative to the rigid case. Porosity, on the other hand, is shown to be more effective in reducing the radiated sound for low k₀. The combined beneficial effects of elasticity and porosity are shown to be effective in reducing the scattered sound for a broader range of k₀ for perforated elastic plates.

1. Introduction

The scattering of surface pressure fluctuations by solid structures produces undesirable noise in a variety of aeroacoustic applications such as trailing-edge noise generation. Recent theoretical work [1] motivated by the silent flight of owls suggests that the noise scattered by an edge can be mitigated by the application of edge elasticity and porosity. In certain parameter ranges identified in their analysis, Jaworski & Peake [1] predict that noise from turbulence scattering by a trailing edge in a low-Mach-number flow can be effectively eliminated, in a scaling sense, by reducing its sound levels to the same order as or weaker than secondary noise sources, such as roughness noise [2,3].

However, some assumptions of the theoretical analysis, such as the semi-infinite extent of the perforated elastic edge that is valid for high frequencies, may need to be relaxed for practical applications. The issue of leading-edge backscattering of trailing-edge sound from a rigid plate has been investigated analytically by Howe [4] and Roger & Moreau [5] for scenarios with and without a background mean flow, respectively. Compliant finite edges are susceptible to resonance effects [6–8] that present practical issues related to the installation of such edge extensions, including the efficacy of the edge to reduce noise levels in a desired frequency range. Other more fundamental concerns include the possibility of secondary scattering of turbulence and elastic waves by the rigid–elastic juncture [9], which may affect the overall far-field acoustic signature.

To address these issues, the present investigation employs a boundary element method (BEM) formulation to solve for the acoustic field due to a turbulent source in proximity to a finite elastic plate with perforations. The goals of this effort are to extend beyond the analytical work of Jaworski & Peake [1] to evaluate finite length effects and to develop a more general computational framework to handle the three-dimensional scattering of compact and non-compact sources. For such, we use the orthonormal basis of free-vibration structural modes to formulate the boundary conditions of the acoustic problem. Other choices of orthogonal modes would also be possible; free-vibration modes are chosen here because they can be obtained with relative ease. A similar approach has been applied by Leclaire et al. [10,11] for the investigation of the vibration problem in porous plates; here, the BEM, is used in order to study acoustic scattering of sources near an edge, expected to be the dominant sound–source mechanism in the aeroacoustics of aerofoils and wings [12].

The remainder of the paper is outlined as follows. First, §2a outlines the model equations for the acoustic scattering problem and the vibration of the perforated elastic plate, including their dimensionless forms. Section 2b derives the solution for the finite elastic plate in terms of a modal basis of structural modes, which are linked to the fluid dynamics via the acoustic pressure. In §2c, the problem is simplified to the case of plates of infinite span but finite chord, which are the focus of this work. Section 3a elaborates on the boundary element method that solves the acoustic problem subjected to boundary conditions related to the vibration of the plate and §3b presents the pseudo-spectral method for the structural eigenvalue problem. Results and discussion are presented in §4, where numerical solutions for elastic impermeable and rigid porous plates are presented in §4a and 4b, respectively, and perforated elastic plates are considered in §4c. The paper is closed by the presentation of conclusions in §5.

2. Mathematical model

(a). Basic equations

The model problem at hand is shown schematically in figure 1. A sound source S, representative of a turbulent eddy in the plate boundary layer, is placed on the vicinity of one of the edges of a finite perforated elastic plate, and we wish to determine the scattered sound at a given observer position r. Figure 1a illustrates a plate with infinite span and finite chord, with a clamped leading edge and free trailing edge, but sound scattered by other combinations of structural boundary conditions can also be handled with the numerical formulation described herein.

Figure 1.

(a) Schematic of a rectangular perforated elastic plate with finite chord and infinite span, where one edge is clamped along the z-axis and the other edge is free, subject to acoustic radiation from source S; (b) cross section, at z=0, illustrates the main physical features of the model problem.

The main physical processes in the problem are represented in figure 1b, which shows a side view of the configuration analysed. The turbulent eddy in the vicinity of the trailing edge generates an incident quadrupolar sound field. A quadrupole in free field has a near-field pressure that is mostly reactive and does not propagate to the far acoustic field. However, owing to the source proximity to the edge, the said near-field pressure is now scattered by the plate and radiates to the far field by this mechanism [13]. The scattered sound radiates as a more efficient dipolar or cardioid sound field, increasing the far-field noise. At the same time, the incident acoustic field excites structural bending waves along the plate that propagate along the surface. These waves hit the clamped leading edge of the plate and are reflected towards the trailing edge. Furthermore, a secondary acoustic scattering takes place at the leading edge of the plate owing to acoustic diffraction and owing to the impingement of the bending waves.

To obtain the scattered sound, we solve the Helmholtz equation,

$${\mathrm{\nabla }}^2\tilde{p}+{\tilde{k}}_0^2\tilde{p}=-\tilde{S},$$ 2.1

where $\tilde{S}$ is the acoustic source function, ${\tilde{k}}_0$ is the acoustic wavenumber, given as $\tilde{\omega }/{\tilde{c}}_0$ for angular frequency $\tilde{\omega }$ and speed of sound ${\tilde{c}}_0$. The overhead tildes indicate dimensional terms, and an $\exp (-\mathrm{i}\tilde{\omega }\tilde{t})$ time dependence is assumed throughout. Equation (2.1) is subject to boundary conditions matching velocities of the fluid and vibrating plate at the fluid–solid interface. The equation for a harmonic load applied to a thin perforated elastic plate is [1,14]

$$(1-{\alpha }_{\mathrm{H}})\tilde{B}{\mathrm{\nabla }}^4\tilde{\eta }-\tilde{m}{\tilde{\omega }}^2\tilde{\eta }=\left(1+\frac{2{\alpha }_{\mathrm{H}}{\tilde{K}}_{\mathrm{R}}}{\pi \tilde{R}}\right)\mathrm{\Delta }\tilde{p},$$ 2.2

where $\tilde{\eta }$ is the plate displacement, $\tilde{B}$ is the effective bending stiffness of the plate (modified by porosity), $\tilde{m}$ is the mass per unit area and $\mathrm{\Delta }\tilde{p}$ is the applied pressure load in the positive $\tilde{y}$-direction. The thin-plate model assumes that the bending wavelength is much longer than the plate thickness. The porosity of the plate is characterized by the open area fraction α_H, the Rayleigh conductivity ${\tilde{K}}_{\mathrm{R}}$ and the pore radius $\tilde{R}$. Note that (2.2) is valid for ${\alpha }_{\mathrm{H}}^2\ll 1$ and ${\tilde{k}}_{0}R\ll 1$ and is the result of area averaging of the plate and the pores, where the local details of individual pores are neglected in favour of their overall influence [14]. $\tilde{B}$ is defined by $\tilde{B}=(1-2{\alpha }_{\mathrm{H}}\nu /(1-\nu )){\tilde{B}}_0$, where ${\tilde{B}}_0$ is the bending stiffness of a plate without porosity and ν is the Poisson ratio.

The Rayleigh conductivity relates ${\tilde{\eta }}_a$, the fluid displacement in the pores, to $\mathrm{\Delta }\tilde{p}$ by

$${\tilde{\eta }}_a=-\frac{{\tilde{K}}_{\mathrm{R}}\mathrm{\Delta }\tilde{p}}{\pi {\tilde{\rho }}_{\mathrm{f}}{\tilde{\omega }}^2{\tilde{R}}^2},$$ 2.3

where ${\tilde{\rho }}_{\mathrm{f}}$ is the fluid density. Finally, the pressure at the plate surface and the plate and fluid displacements are related by the linearized Euler equation as

$${\tilde{\rho }}_{\mathrm{f}}{\tilde{\omega }}^2[(1-{\alpha }_{\mathrm{H}})\tilde{\eta }+{\alpha }_{\mathrm{H}}{\tilde{\eta }}_a]={\left.\frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }\tilde{y}}\right|}_{\tilde{y}=0}.$$ 2.4

Following Jaworski & Peake [1] and Crighton & Innes [15], proceed to obtain non-dimensional equations by defining the coincidence frequency,

$${\tilde{\omega }}_{\mathrm{c}}={\left(\frac{\tilde{m}{\tilde{c}}_0^4}{\tilde{B}}\right)}^{1/2},$$ 2.5

the vacuum bending wave Mach number,

$$\mathrm{\Omega }={\left(\frac{\tilde{\omega }}{{\tilde{\omega }}_{\mathrm{c}}}\right)}^{1/2}=\frac{{\tilde{k}}_0}{{\tilde{k}}_B},$$ 2.6

and the intrinsic fluid-loading parameter,

$$ϵ=\frac{{\tilde{\rho }}_{\mathrm{f}}{\tilde{k}}_0}{\tilde{m}{\tilde{k}}_B^2}=\frac{{\tilde{\rho }}_{\mathrm{f}}{\tilde{c}}_1}{{\tilde{\rho }}_s{\tilde{c}}_0}\sqrt{\frac{1-2\nu }{12{(1-\nu )}^2}}.$$ 2.7

In the equations above, ${\tilde{k}}_B$ is the bending wavenumber, ${\tilde{c}}_1$ is the speed of longitudinal compression waves in the solid and ${\tilde{\rho }}_s$ is the mass density of the material of the plate, related to the mass per unit area as ${\tilde{\rho }}_s=\tilde{m}/\tilde{h}$. Here, $\tilde{h}$ is the thickness of the plate. The speed of longitudinal waves ${\tilde{c}}_1$ is here used simply to define the non-dimensional fluid-loading parameter; in the present structural model, only flexural waves are considered via the bending-wave equation (2.2). Note that the fluid-loading parameter ϵ depends solely on properties of the fluid and the plate [14].

The non-dimensional Rayleigh conductivity, $K_{\mathrm{R}}=2{\tilde{K}}_{\mathrm{R}}/(\pi \tilde{R})$, is obtained for circular apertures where ${\tilde{K}}_{\mathrm{R}}=2\tilde{R}$, and thus K_R=4/π. Extensions of the Rayleigh conductivity concept to non-circular orifices and external flow effects may be considered by the methods described by [16,17] but are not pursued here.

After identifying the following dimensionless variables,

$$\left.\begin{array}{rl}\eta & =\frac{\tilde{\eta }}{\tilde{L}},R=\frac{\tilde{R}}{\tilde{L}},k_0={\tilde{k}}_0\tilde{L}\text{and}\\ p& =\frac{\tilde{p}}{({\tilde{\rho }}_{\mathrm{f}}{\tilde{c}}_0^2)},S=\frac{\tilde{S}}{({\tilde{\rho }}_{\mathrm{f}}{\tilde{c}}_0^2{\tilde{L}}^2)},(x,y,z)=\frac{(\tilde{x},\tilde{y},\tilde{z})}{\tilde{L}},\end{array}\right\}$$ 2.8

where $\tilde{L}$ is the reference chord length of the finite perforated elastic plate, we arrive at the non-dimensional versions of equations (2.1)–(2.4):

$$\begin{array}{rl}& {\mathrm{\nabla }}^{2}p+k_0^{2}p=-S,\end{array}$$ 2.9

$$\begin{array}{rl}& (1-{\alpha }_{\mathrm{H}}){\mathrm{\nabla }}^4\eta -\frac{k_0^4}{{\mathrm{\Omega }}^4}\eta =(1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}})\frac{ϵ}{{\mathrm{\Omega }}^6}{k}_0^3\mathrm{\Delta }p\end{array}$$ 2.10

$$\begin{array}{rl}\text{and}& (1-{\alpha }_{\mathrm{H}})k_0^2\eta -\frac{{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{2R}\mathrm{\Delta }p={\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\right|}_{y=0}.\end{array}$$ 2.11

The system of equations (2.9)–(2.11) constitute the acoustic problem (2.9) subject to boundary conditions (2.10) and (2.11) that relate the pressure and its normal derivative on the plate surface. For a given set of plate parameters, the structural–acoustic interaction is governed by three non-dimensional parameters: ϵ, Ω and k₀. To close the problem, the boundary conditions of the vibration problem (2.10) must be provided, where only two such conditions may be specified at each end of the plate.

(b). Solution of problem using a structural modal basis

We rewrite equation (2.10) as

$$(1-{\alpha }_{\mathrm{H}})\mathcal{L}(\eta )-\frac{k_0^4}{{\mathrm{\Omega }}^4}\eta =(1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}})\frac{ϵ}{{\mathrm{\Omega }}^6}{k}_0^3\mathrm{\Delta }p,$$ 2.12

where $\mathcal{L}={\mathrm{\nabla }}^4$, which is subject to homogeneous boundary conditions. Consider now the eigenvalue problem $\mathcal{L}(\eta )={\beta }^4\eta$ subject to the same boundary conditions. Solutions of this eigenvalue problem form a complete orthonormal basis ϕ_i for functions satisfying the boundary conditions of the problem [18,19], such that

$$\mathcal{L}({\varphi }_i)={\beta }_i^4{\varphi }_i,\text{where}\hspace{0.17em}⟨{\varphi }_i,{\varphi }_j⟩={\delta }_{ij}.$$ 2.13

In what follows, we shall call ϕ_i the modal basis. The eigenvalues of this problem are real and positive; the convention β⁴ is thus justified and allows the identification of β_i as the bending wavenumber of a vibration mode ϕ_i of the plate. This modal basis comprises the in vacuo, free-vibration modes of the plate. In what follows, we will use these modes as an auxiliary basis to solve the fluid-loaded plate problem, where the plate is saturated with fluid, and the pressure is coupled to the plate and fluid displacements. For cases with light fluid loading (ϵ≪1), the free-vibration modes are expected to be close to those for fluid-loaded plates, and the convergence of the plate displacement in terms of free-vibration modes is expected to be rapid.

Because the modal basis is a complete orthonormal set for functions satisfying the boundary conditions of the problem, we can write

$$\eta =\sum _i{a}_i{\varphi }_i$$ 2.14

for any solution η of the problem composed of (2.12) and the associated boundary conditions. The coefficients a_i are determined by substituting (2.14) into (2.12), using (2.13), and then taking the inner product with ϕ_j. Thus, the displacement (2.14) is

$$\eta =\frac{1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{1-{\alpha }_{\mathrm{H}}}\frac{ϵk_0^3}{{\mathrm{\Omega }}^6}\sum _j\left[\frac{⟨\mathrm{\Delta }p,{\varphi }_j⟩}{{\beta }_j^4-k_0^4/[(1-{\alpha }_{\mathrm{H}}){\mathrm{\Omega }}^4]}{\varphi }_j\right],$$ 2.15

and the derivative of the pressure in the transverse direction evaluated at the plate surface (2.11) is

$${\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\right|}_{y=0}=(1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}})\frac{ϵk_0^5}{{\mathrm{\Omega }}^6}\sum _j\left[\frac{⟨\mathrm{\Delta }p,{\varphi }_j⟩}{{\beta }_j^4-k_0^4/[(1-{\alpha }_{\mathrm{H}}){\mathrm{\Omega }}^4]}{\varphi }_j\right]-\frac{{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{2R}\mathrm{\Delta }p,$$ 2.16

where the term 〈Δp,ϕ_j〉 is the inner product ${\int }_0^1\mathrm{\Delta }p(x){\varphi }_j(x)\hspace{0.17em}\mathrm{d}x$.

Equation (2.16) relates the pressure difference between the two sides of the plate Δp with the transverse pressure gradient evaluated at the plate surface ∂p/∂y|_y=0 by the solution of the vibration problem. The acoustic problem, formulated with a boundary element method, is based on an integral equation where p and ∂p/∂n|_Γ are to be solved for; thus, (2.16) couples these two quantities through the vibration of the plate. The particular case of an impermeable elastic plate can be obtained by setting α_H=0 in (2.16); similarly, results for a porous rigid plate can be obtained by setting ϵ=0 in (2.16).

The modal expansion is written using M modes. We observe in (2.16) that the important modes will be the ones for which β_j≈k_B, which will happen when the plate is excited near a resonance frequency. We expect only limited contributions of modes that are far from satisfying this condition, and thus the truncation of the expansion is chosen such that β_M, the eigenvalue of the last mode in the expansion, is significantly higher than the bending wavenumber k_B.

(c). Plates with infinite span

The formulation in §2b applies generally to rectangular plates. In the following sections, we will now deal solely with the two-dimensional acoustic scattering from a plate of infinite span. We consider a clamped leading edge at x=0 and a free trailing edge at x=1. In this case, there is no z-dependence for all variables, and the boundary conditions are

$$\eta (0)={\eta }^{\prime }(0)={\eta }^{″}(1)={\eta }^{‴}(1)=0,$$ 2.17

where ( )′=∂/∂x and the $\mathcal{L}$ operator reduces to ∂⁴/∂x⁴.

3. Numerical methods

(a). Boundary element formulation

We now develop the boundary element scheme to solve the problem of acoustic scattering by a finite perforated elastic plate. The sound waves are produced by concentrated sources representative of turbulent eddies positioned near the plate trailing edge. The following non-homogeneous Helmholtz equation represents the pressure disturbances induced by the concentrated sources in a quiescent medium,

$${\mathrm{\nabla }}^{2}p(\mathbf{x})+k_0^{2}p(\mathbf{x})=-S_i.$$ 3.1

In (3.1), S_i represents the ith source strength, and all terms are written as non-dimensional quantities following the procedure discussed previously. A quadrupole source is chosen to provide the incident acoustic field because it represents the noise from a compact turbulent eddy [20,12]. A fundamental solution for the Helmholtz equation is the free space Green's function, $G(\mathbf{x},\mathbf{y})$, written as

$$G(\mathbf{x},\mathbf{y})=\frac{i}{4}{H}_0^{(1)}(k_0|\mathbf{x}-\mathbf{y}|),$$ 3.2

for a two-dimensional formulation. Here, $H_0^{(1)}$ stands for the Hänkel function of the first kind and order zero. The incident quadrupolar field can be computed as the second derivative of the Green's function.

Using Green's second identity, one can write the following boundary integral equation

$$T(\mathbf{x})p(\mathbf{x})={\int }_{\mathrm{\Gamma }}\left[\frac{\mathrm{\partial }p(\mathbf{y})}{\mathrm{\partial }{n}_y}G(\mathbf{x},\mathbf{y})-\frac{\mathrm{\partial }G(\mathbf{x},\mathbf{y})}{\mathrm{\partial }{n}_y}p(\mathbf{y})\right]\mathrm{d}\mathrm{\Gamma }-\frac{{\mathrm{\partial }}^{2}G(\mathbf{x},{\mathbf{z}}_i)}{\mathrm{\partial }{\mathbf{z}}_{i_m}\mathrm{\partial }{\mathbf{z}}_{i_n}}S({\mathbf{z}}_i),$$ 3.3

where $T(\mathbf{x})=\frac{1}{2}$ when $\mathbf{x}$ is on a smooth boundary surface Γ, and $T(\mathbf{x})=1$ when $\mathbf{x}$ is a field point anywhere in the fluid region. The derivatives with respect to the inward normal direction of the boundary surface are represented by ∂/∂n and $\mathbf{n}$ is an inward unit normal. The ith source location is ${\mathbf{z}}_i$.

In the BEM formulation, the scattering surface, Γ, is discretized into a finite number of elements with polynomial reconstructions for the unknowns in each element. Then, (3.3) is solved for each of these elements through the solution of a dense linear system of equations such as

$$[H]\{p\}-[G]\left\{\frac{\mathrm{\partial }p}{\mathrm{\partial }n}\right\}=\{S\}.$$ 3.4

The coefficients of matrix [H] are given by $h_{ij}=\frac{1}{2}$ for i=j and $h_{ij}={\int }_{\mathrm{\Gamma }}(\mathrm{\partial }G({\mathbf{x}}_i,{\mathbf{x}}_j)/\mathrm{\partial }{n}_j)\hspace{0.17em}\mathrm{d}\mathrm{\Gamma }$ for i≠j. The coefficients of matrix G are written as $g_{ij}={\int }_{\mathrm{\Gamma }}G({\mathbf{x}}_i,{\mathbf{x}}_j)\hspace{0.17em}\mathrm{d}\mathrm{\Gamma }$. The boundary conditions specified on the surface of the plate are calculated using the derivative of the pressure in the transverse direction,

$${\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }n}\right|}_{\mathrm{\Gamma }}=n_y{\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\right|}_{y=0}=n_y(1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}})\frac{ϵk_0^5}{{\mathrm{\Omega }}^6}\frac{⟨\mathrm{\Delta }p(x),{\varphi }_j⟩}{{\beta }_j^4-k_0^4/[(1-{\alpha }_{\mathrm{H}}){\mathrm{\Omega }}^4]}{\varphi }_j-n_y\frac{{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{2R}\mathrm{\Delta }p.$$ 3.5

One can rewrite (3.4) in terms of the acoustic pressure and obtain a direct solution of the coupled problem. Hence, the new linear system is given by

$$([H]-[G][D])\{p\}=\{S\},$$ 3.6

where the individual elements composing the D matrix are written for the discrete boundary elements as

$$\begin{array}{rl}{D}_{i,k}& =(1+{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}})\frac{ϵk_0^5}{{\mathrm{\Omega }}^6}{n}_{y_i}{n}_{y_k}{\gamma }_k\sum _{j=1}^M\frac{\varphi {(x_k)}_j\varphi {(x_i)}_j}{{\beta }_j^4-k_0^4/[(1-{\alpha }_{\mathrm{H}}){\mathrm{\Omega }}^4]}\\ & -\frac{{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{2R}{n}_{y_i}(n_{y_k}{\delta }_{k,i}+n_{y_k}{\delta }_{k,N-i}).\end{array}$$ 3.7

In equation (3.7), N is the number of boundary elements used in the plate discretization and γ is the length of a particular boundary element. We are also assuming here that boundary elements i and N−i have centroids at identical x positions, i.e. if element i is on the top (bottom) of the plate, element N−i is at the same x position in the bottom (top) of the plate. The linear system in (3.6) is solved using standard Gaussian elimination.

(b). Vibration problem

The eigenvalue problem (2.13) is solved using a pseudo-spectral method (or spectral collocation) [21,22]. The motivation for using this method is the higher accuracy compared with other numerical schemes such as the finite difference method. In particular, for low Ω, a high number of vibration modes are needed to obtain the solution of the coupled fluid–structure problem, and the spectral convergence eases the obtention of a high number of eigenvalues and eigenfunctions [22]. Furthermore, the current approach can be easily adapted for use in the solution of the three-dimensional problem. Here, we define auxiliary variables

$$w_1=\eta (x)\text{and}{w}_2={\eta }^{″}(x)$$ 3.8

to obtain the eigenvalue problem in matrix form:

$$\left[\begin{array}{cc}{\mathcal{D}}^2& -\mathcal{I}\\ 0& {\mathcal{D}}^2\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]={\beta }^4\left[\begin{array}{cc}0& 0\\ \mathcal{I}& 0\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right],$$ 3.9

where $\mathcal{D}=\mathrm{\partial }/\mathrm{\partial }x$ and $\mathcal{I}$ is the identity operator.

The derivative operator is written in matrix form by a discretization using Chebyshev polynomials [21]. Boundary conditions are imposed in (3.9) by replacing corresponding lines of the matrix in the left-hand side with either $\mathcal{I}$ or $\mathcal{D}$, and lines of the right-hand side are zeroed. Boundary conditions for a clamped edge are imposed on w₁, and for a free edge on w₂.

For the vibration problem, we employ a discretization using 321 Chebyshev polynomials. The numerical solution gives values of the modes on the Chebyshev grid; the modes are then sampled using barycentric interpolation [23] at a grid x_i of N points, with locations chosen to be appropriate for the application of the boundary element method for the acoustic problem.

4. Results and discussion

To study an acoustic scattering problem relevant for aeroacoustics, we have calculated the sound radiated by the free edge of a perforated, elastic plate near a lateral point quadrupole source of unit intensity. The free edge is located at (x,y)=(1,0), and the quadrupole source is placed at (x,y)=(1,0.004). We calculate the relative change in acoustic power level due to effects of porosity and elasticity and present directivity results for observers in the acoustic far-field located 50 chords from the plate trailing edge; the polar angle θ is measured according to the schematic in figure 1b. A comparison in terms of acoustic results obtained by the boundary element formulation is presented against semi-infinite plate results from the literature in the appendix A.

To use the present formulation of the BEM code, it is necessary to calculate the radiated sound by a plate of small but finite thickness. In this work, we have worked with a plate with thickness h equal to 0.2% of its chord. In all cases analysed in this work, this thickness is much smaller than the acoustic wavelength and, therefore, the directivities obtained by the BEM are close to the expected results for plates of zero thickness [24]. Unless otherwise noted, we have chosen ϵ=0.0021 as representative of an aluminium plate in air [14] whenever elasticity effects are considered. One hundred in vacuo bending modes are used for all simulations in the manuscript. For all cases analysed, the frequency of the 100th mode was considerably above the structural frequencies excited. We have done verifications for the extreme cases considered in this work (high k₀ and low Ω, leading to high k_B) by changing the number of bending modes to 150; resulting directivities were nearly identical. A mesh convergence study was also performed and numerical simulations are conducted using 802 boundary elements in the plate discretization. Whenever possible we present comparisons with corresponding results for semi-infinite plates [1], which also serve as validation for our method; however, an exact agreement should not be expected owing to leading-edge backscattering of acoustic [4,5] and bending waves [15].

(a). Impermeable elastic plate

The case of a non-porous elastic plate corresponds to α_H=0 in the present formulation, and (2.16) becomes

$${\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\right|}_{y=0}=\frac{ϵk_0^5}{{\mathrm{\Omega }}^6}\frac{⟨\mathrm{\Delta }p,{\varphi }_j⟩}{{\beta }_j^4-k_0^4/{\mathrm{\Omega }}^4}{\varphi }_j.$$ 4.1

For all cases analysed, the Helmholtz number is defined by ${\tilde{k}}_0\tilde{L}=k_0,$ where $\tilde{L}$ is the plate chord. A parametric study has been performed so as to evaluate the combined effects of elasticity and finiteness of the plate; the latter is a function of k₀, where k₀=0.1 is taken as representative of the compact surface limit (the plate chord is much smaller than the acoustic wavelength), and k₀=10 tends to the non-compact limit (the chord becomes larger than the acoustic wavelength).

It is apparent from (4.1) that the rigid-plate limit ∂p/∂y=0 is recovered when ϵ=0, Ω≫1 or k₀≪1. We hence anticipate that finite-plate acoustic effects will lead to reduced changes when compared with the rigid limit; in particular, acoustically compact elastic plates for which k₀≪1 should radiate similarly to rigid ones, unless Ω≪1 or close to a structural resonance, where β_j≈k₀/Ω for some j. These features will be seen in the results of this section.

To show the general changes of the radiated sound owing to plate elasticity, figure 2 displays sample pressure fields for an aluminium plate immersed in water, which corresponds to a higher fluid loading (ϵ=0.135) than what is considered elsewhere in this paper; this choice was made for the ease of visualization of the relevant waves in the problem.

Figure 2.

Sample scattered pressure fields for an aluminium plate immersed in water (ϵ=0.135), with k₀=20: (a) rigid plate, (b) Ω=0.3 and (c) Ω=0.1. Figures are plotted using the same contour levels for acoustic pressure. The plate thickness is exaggerated in the figures for ease of visualization. (Online version in colour.)

We consider first the scattering of an acoustic quadrupole in the vicinity of the trailing edge of a rigid flat plate, shown in figure 2a. We see that in the vicinity of the trailing edge (x,y)=(1,0) the scattered field presents the cardioid directivity typical of the semi-infinite edge problem [12], with waves with maximal amplitude propagating towards the plate leading edge. As these acoustic waves approach the leading edge, they are backscattered; this process occurs multiple times and leads to a directivity shape close to the original cardioid, modified by lobes [25,26]. These lobes are visible in figure 2a, especially near the leading edge.

Fluid-loaded elastic plates are shown in figure 2b,c. In addition to the acoustic scattering discussed above for elastic plates, the quadrupole incident field is also scattered into bending waves, which are subsonic (Ω=0.3 and 0.1 for figure 2b and c, respectively) and, hence, only lead to evanescent pressure waves, decaying exponentially in |y|; these are visible in the immediate neighbourhood of the flat plate. Once these bending waves reach the trailing edge, they are reflected, leading to a standing-wave pattern, but are also scattered as acoustic waves. This is visible near the leading edge of figure 2b,c. If the plate elasticity is high, which here corresponds to a low Ω, all the scattered waves discussed here have their amplitudes reduced, as shown in figure 2c.

To evaluate the reduction of acoustic scattering with plate elasticity, figure 3 shows the change of sound power level (ΔPWL) relative to the rigid limit as a function of Ω and k_B for plates with Helmholtz numbers ranging from 0.1 to 1, considering an aluminium plate immersed in air (ϵ=0.0021). A first apparent feature of figure 3a,b is the presence of sharp peaks, which correspond to resonances of the fluid-loaded plates. These resonances are more clearly seen by comparing k_B for the peaks with the in vacuo resonance wavenumbers β_i, shown with arrows in figure 3b. We observe that each resonance occurs for slightly lower k_B than the corresponding in vacuo value, a feature expected theoretically [15] and observed experimentally [27]. These differences between in vacuo and fluid-loading resonances are more pronounced for lower k₀.

Figure 3.

Change in power level radiated by elastic plates relative to the rigid limit, as a function of (a) bending wave Mach number Ω and (b) bending wavenumber k_B. Vertical arrows in (b) show resonance wavenumbers for a clamped-free plate in vacuo. (Online version in colour.)

As k_B is increased (or Ω is reduced) the acoustic excitation frequency crosses successive resonance conditions. For each resonance crossing, the phase between excitation and plate vibration near the trailing edge jumps by a value of π, which can be seen as a change of sign of the denominator in (4.1). This phase jump leads to increases and decreases of PWL at either side of each structural resonance. We observe that between resonance numbers 2n−1 and 2n, where n=1,2,3…, there are more significant decreases of PWL relative to the rigid limit when compared to the cases between resonance numbers 2n and 2n+1.

Excluding the specific values corresponding to plate resonance, the results in figure 3a demonstrate more significant reductions of PWL as the bending-wave Mach number Ω is decreased, which is consistent with results for semi-infinite plates [28]. However, finite-plate acoustic effects, manifested here for k₀<1, lead to lower reduction of PWL if compared with the semi-infinite limit; this can be seen by noting the lower absolute values of k₀=0.1 and 0.3 compared with k₀=1. Therefore, in the case of an acoustically compact chord, the aggregate effect of acoustic and elastic wave scattering by the leading edge is to diminish the noise reduction capacity of elastic trailing edges.

The ΔPWL results are extended for higher k₀ in figure 4, where results for k₀=1 are repeated for reference. Because k_B=k₀/Ω, results plotted as a function of Ω present a higher density of resonance peaks as k₀ is increased, which explains the general behaviour of figure 4. Besides this feature, for increasing values of k₀>1, we observe a convergent trend of ΔPWL values between the resonance peaks towards the non-compact limit of a semi-infinite plate [1].

Figure 4.

Change in power level radiated by elastic plates relative to the rigid limit as a function of bending wave Mach number Ω. (Online version in colour.)

Representative compact and non-compact directivity patterns are shown for k₀=0.1 and k₀=10.0 in figure 5a and b, respectively. In the rigid limits, we observe the typical compact dipole behaviour for k₀=0.1, and the cardioid directivity shape modified by the presence of lobes for k₀=10. Reductions of plate stiffness lead to lower Ω and, in both cases, to weaker acoustic scattering. We observe that for lower k₀ the directivity shape is preserved; this was also seen for k₀=0.3 and 1 (not shown here). For higher k₀, the cardioid shape switches progressively with increasing k_B to that of a compact dipole, as seen in figure 5b for Ω=0.06. This trend is consistent with what is observed for semi-infinite plates (see [28] and appendix). The comparison between k₀=0.1 and 10 also highlights that, for k₀<1, lower Ω is required for significant reductions of acoustic scattering when compared with k₀=10.

Figure 5.

Amplitude (|p′|) of scattered far-field pressure for an impermeable, elastic plate for varying Ω=(ω/ω_c)^1/2=k₀/k_B and Helmholtz number (a) k₀=0.1 and (b) k₀=10. (Online version in colour.)

(b). Perforated rigid plate

The case of a rigid plate with perforation corresponds to ϵ=0 in the present formulation, and (2.16) becomes

$${\left.\frac{\mathrm{\partial }p}{\mathrm{\partial }y}\right|}_{y=0}=-\frac{{\alpha }_{\mathrm{H}}{K}_{\mathrm{R}}}{2R}\mathrm{\Delta }p.$$ 4.2

By inspection, (4.2) indicates that ∂p/∂y|_y=0 is always in phase opposition with Δp; this phase opposition should diminish the radiated sound. Equation (4.2) also shows that the effect of porosity can be evaluated by looking solely at the non-dimensional group α_HK_R/2R, as in Ffowcs Williams [29] and Jaworski & Peake [1]; accordingly, we expect more significant reductions of the radiated sound as α_HK_R/2R is increased. In what follows, we take K_R=4/π and evaluate the effect of α_H/R and μ/k₀=α_HK_R/(k₀R) on the radiated sound for a range of Helmholtz numbers. Results for these two parametric variations are shown in figure 6.

Figure 6.

Reduction of power level (PWL) of perforated plates compared with the impermeable limit, shown as a function of (a) α_H/R and (b) μ/k₀=α_HK_R/(k₀R). Straight dashed lines show the (α_H/R)⁻¹ slope in (a), and the non-compact, high μ/k₀ limit ΔPWL=k/(2μ) in (b). (Online version in colour.)

Unlike the elasticity effect, which for given Ω leads to highest PWL reductions for large Helmholtz numbers, the porosity effect leads to more significant reductions of the scattered sound for lower k₀, as shown in figure 6a; however, the trend is not monotonic, with k₀=1 presenting higher ΔPWL (in absolute value) than compact plates with k₀=0.1 and 0.3. For all plates, two behaviours are observed: in the small porosity limit, α_H/R≪1 changes in acoustic scattering become negligible, and for high α_H/R, the resulting ΔPWL is proportional to (α_H/R)⁻¹.

The reduction of PWL by semi-infinite perforated plates is given as ΔPWL=k/(2μ) for μ≫k₀ [1,30]. This limit, as well as the corresponding results for semi-infinite plates for arbitrary μ/k₀, are shown in figure 6b. We see that the present results for k₀=5 and 10 are close to the semi-infinite limit; in these cases, the effect of k₀ can be absorbed into the non-dimensional parameter μ/k₀, and the leading-edge effect can be subsequently neglected.

Figure 7 presents directivity patterns for compact and non-compact perforated plates, where porosity leads to reductions of the radiated sound in all directions. For lower k₀, illustrated by the k₀=0.1 results in figure 7a, the directivity maintains its compact dipole shape, decreasing in amplitude with increasing α_H/R; and for higher k₀ (10 for figure 7b,) the directivity shape changes progressively from a lobed cardioid to the dipole shape for higher α_H/R. As discussed previously, there is a Helmholtz number effect noted by a comparison between figure 7a and b, where the effect of porosity on sound reduction is diminished as k₀ increases. This trend is the inverse of what was observed for elasticity in §4a, where lower reductions (and in some cases increases) of the sound radiation are obtained for small k₀. These observations suggest that a combination of elasticity and porosity may be a viable way to reduce scattered sound for both lower k₀ (where porosity is more efficient in reducing scattering) and higher k₀ (where elasticity leads to significant noise reductions compared with the rigid case). Perforated elastic plates are examined in §4c.

Figure 7.

Amplitude (|p′|) of far-field pressure for porous rigid plate with (a) k₀=0.1 and (b) k₀=10. (Online version in colour.)

(c). Perforated elastic plates

For the problem of scattering by perforated elastic plates, whose boundary conditions are obtained using the full version of (2.16), the radiated sound depends on the Helmholtz number k₀, the fluid–structure parameters ϵ, Ω and the porosity parameters α_H, k_R and R. A broad parametric study would therefore be an exhaustive task. In this section, we have chosen to fix R=10⁻³ and to study α_H/R=0.5 and 2.0 with Ω sweeps.

Figure 8 shows values of ΔPWL for perforated elastic plates computed relative to the rigid, impermeable limit; as such, each case contains the combined effect of elasticity and perforation. Figure 8a,b for the perforated elastic plate is directly comparable to the strictly elastic plate results in figure 3a and display similar trends, with peaks appearing owing to structural resonance. The main difference between these figures is that for the highest Ω displayed there are already reductions of the radiated power level for the perforated elastic case, which are entirely owing to the perforations (a comparison with the results of figure 6 shows that the Ω=0.5 values of ΔPWL are the same as those for perforated rigid plates). As the bending stiffness is reduced (lower Ω), we obtain combined effects of elasticity and porosity, and further reductions of the radiated power are obtained.

Figure 8.

Change in power level radiated by perforated elastic plates with k₀ ranging from 0.1 to 1, relative to the rigid impermeable limit, for R=10⁻³ and (a) α_H/R=0.5 and (b) α_H/R=2.0. (Online version in colour.)

Similar ΔPWL plots are shown in figure 9, this time for k₀=1, 5 and 10; these results are directly comparable to figure 4. Low Ω results for k₀=10 were not plotted to avoid clutter in the plots. As discussed in §4b, for higher k₀, the effect of perforations in the radiated sound is reduced and, hence, the values of ΔPWL are closer to those obtained for impermeable elastic plates, especially for k₀=10. The trend in Ω observed here is thus similar to what is seen in figure 4a. For the present set of parameters, the greatest reductions of scattered sound are obtained for k₀=1, which is an intermediate value of the Helmholtz number where elasticity and porosity have a significant effect in reducing the radiated sound compared with the rigid, impermeable limit.

Figure 9.

Change in power level radiated by perforated elastic plates with k₀ ranging from 1 to 10, relative to the rigid impermeable limit, for R=10⁻³ and (a) α_H/R=0.5 and (b) α_H/R=2.0. (Online version in colour.)

Directivity results are shown in figure 10, where the same parameters of figure 5 are used to allow comparison between the sound field of impermeable and perforated elastic plates. The same behaviours found in the preceding sections are seen here: the directivity shape remains in a compact dipole shape for k₀=0.1, with lower amplitudes depending on elasticity and porosity, whereas the reductions of radiated sound for higher Helmholtz number (illustrated here with k₀=10) change the directivity shape from a cardioid shape to a compact dipole.

Figure 10.

Amplitude (|p′|) of scattered far-field pressure from a perforated, elastic plate for R=10⁻³, α_H/R=2.0, varying Ω and Helmholtz number (a) k₀=0.1 and (b) k₀=10. (Online version in colour.)

The overall behaviours observed here highlight that the combined effects of elasticity and porosity lead to more significant reductions of the radiated sound. For higher k₀, elasticity plays a more significant role and, for lower k₀, porosity leads to more significant reductions even for cases where elastic effects are negligible.

5. Conclusion

We present a formulation to compute the acoustic field scattered by finite perforated elastic plates. An existing model [14,1] is recast in terms of the modal basis from the associated free-vibration problem. The modal basis enables a direct numerical solution of the fluid–structure interaction using a BEM formulation, with boundary conditions obtained by an expression of ∂p/∂n over the surface as a function of the loading Δp projected onto the free-vibration modes.

The boundary element formulation has the practical advantage of computing the numerical solution of the problem in two steps. The modal basis of the free-vibration problem is obtained first; in this work, the modal basis is calculated numerically with a pseudo-spectral method, but other techniques could also be employed. In a second step, the acoustic problem is solved by a boundary element solver, which uses the previously obtained modal basis to relate ∂p/∂n on the plate surface to the pressure difference across the plate. We have focused in the canonical problem of a finite flat plate, but extension of the present formulation for different geometries does not constitute a major difficulty.

With the present method, we can study scattering by finite plates including the effects of elasticity and porosity, which can be treated in isolation or combination. We restrict the analysis to plates with infinite span but finite chord and, therefore, we deal with a two-dimensional acoustic scattering problem. The results demonstrate that, in agreement with the expected trends for semi-infinite plates [1,28,31], elasticity and porosity lead to reductions of the radiated sound when compared with the rigid and impermeable limits, respectively. Each is seen to lead to non-zero values of ∂p/∂y (and thus of ∂p/∂n) over the plate surface, besides the dipoles of intensity Δp already present for rigid, impermeable plates. Porosity alone leads to a phase opposition between ∂p/∂y and Δp and this was related to a reduction of the scattered sound. For elastic plates, there is also scattering of bending waves at the trailing edge, which are reflected by the leading edge and also backscattered as acoustic waves; for low values of the bending-wave Mach number Ω, the whole process leads to reductions of the radiated sound compared with the rigid limit. Unlike the results for semi-infinite plates, structural resonance becomes a relevant phenomenon for the acoustic scattering by finite plates. Large plate displacements are obtained when a plate is excited near the resonance conditions, and this was seen to lead to significant changes to acoustic scattering, with increases or decreases of the radiated sound depending on the phase between acoustic excitation and structural response.

Regarding the obtained reductions of acoustic scattering, we see that finite-plate effects are relevant for Helmholtz number k₀≈1 or lower, with opposing trends for elasticity and porosity. On the one hand, for lower k₀, elastic effects lead to less pronounced reductions of the radiated sound and, in some cases, even an increase of acoustic scattering; for these cases, porosity is seen to be more efficient in reducing the acoustic intensity. On the other hand, for higher values of k₀, elasticity effects reduce noise radiation considerably, as expected from previous investigations of semi-infinite plates. However, porosity effects are almost negligible at higher Helmholtz numbers. When both effects are considered together in the analysis of perforated elastic plates, we observe that significant sound reductions are obtained for all k₀, and this can be understood by noting that the cases where elasticity would play a minor role (lower k₀) are those where porosity is more efficient, and vice versa. Thus, perforated elastic trailing edges appear to be a promising solution to reduce trailing-edge noise in a relatively broad range of frequencies, and the methods presented in this paper allow the evaluation of potential benefits of different porosity and elasticity parameters.

Appendix A. Comparison of directivities with semi-infinite plate results

Here, we compare the results of the present formulation with some results from Jaworski & Peake [1], who considered acoustic scattering by a semi-infinite plate with perforations. The same code of the cited work is used here to generate directivity results that are compared with those of the present formulation. To facilitate a directivity comparison between the present method, a two-dimensional scattering problem must be considered. We emphasize that a direct validation between the semi-infinite model problem with the present computational framework for a finite plate is not possible due to the leading-edge effects on the elastic waves within the plate and the scattered acoustic field. However, we anticipate for large Helmholtz number k₀ that the strength of the scattered sound field from finite plates is consistent with semi-infinite plates, and these comparisons are made here to furnish an indirect validation.

Jaworski & Peake use the reciprocity theorem to calculate acoustic scattering by a point quadrupole in the vicinity of the trailing edge; this is done by solving the reciprocal problem, which consists of finding the near field of the trailing edge subject to an incident spherical wave. In this work, we specify the incident field as a cylindrical wave emitted from a far-field position,

$$p_0=\frac{{\mathrm{e}}^{\mathrm{i}(k_0{r}_0-\pi /4)}}{\sqrt{8\pi k_0{r}_0}},$$ A 1

where the asymptotic form of the Hänkel function for large arguments is used. Using this incident field in the development of Jaworski & Peake, we obtain

$$\left|\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }x\mathrm{\partial }y}\right|=\frac{|B\cos (3\theta /2)|}{\pi r^{3/2}\sqrt{32k_0{r}_0}},$$ A 2

equivalent to their equation (4.21), using the notation of this work. In the present comparison, we have adopted θ=π/2, as in the remainder of the paper (i.e. the compact quadrupole is placed just above the trailing edge). The parameter B is defined by equation (4.16) in [1], which involves the plate edge boundary conditions and Wiener–Hopf kernel factorizations that were carried out numerically and asymptotically by Jaworski & Peake.

For a rigid impermeable plate, $B=\sqrt{2k_0}\sin ({\theta }_0/2)$ [1], which leads to

$$\left|\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }x\mathrm{\partial }y}\right|4\pi r^{3/2}\sqrt{r_0}=\sin \left(\frac{{\theta }_0}{2}\right),$$ A 3

the directivity for the two-dimensional acoustic problem of a quadrupole near the trailing edge of a semi-infinite plate. We compute results of the Wiener–Hopf method from Jaworski & Peake [1] using the same numerical methods in that study, and compare the directivities (obtained by inputting the appropriate value of B in (5.2)) to our numerical results. In all cases, the directivities are rescaled by multiplication to $4\pi r^{3/2}\sqrt{r_0}$ to enable a simple comparison with the rigid limit.

Rigid- and elastic-edge scattering results are shown in figure 11, corresponding to cases presented by Howe [28] for ϵ=0.135; we note the quantitative agreement of the directivity computations by Howe [28] and Jaworski & Peake [1] for semi-infinite elastic plates that was carried out by the authors but is not illustrated here. The present results were generated for k₀=20, with a point quadrupole placed at (x,y)=(1,0.004) and far-field directivity taken at r₀=50. For small bending wave Mach number (cf. Ω²=0.1), we note the relative agreement in magnitude between (5.3) for the rigid semi-infinite plate and the BEM results for the finite rigid plate. Clearly, the presence of the leading edge leads inevitably to a modulation of the directivity, producing a lobed pattern that depends upon Ω. A comparison of elastic edge results from the present finite-plate method against semi-infinite elastic plate results from the Wiener–Hopf approach supports similar conclusions regarding the consistent order-of-magnitude of the scattered sound fields and the modulation of the directivity pattern due to the leading edge. As the bending wave Mach number increases, the elastic results approach their corresponding rigid-edge limits, as expected in the limit of large Ω in (4.1). Finally, we note that the analytical scaling in (5.3) holds for the numerical results, which have the correct overall amplitudes once the scaling is applied.

Figure 11.

Edge noise directivity comparisons for rigid plates and elastic plates with fixed ϵ=0.135. Rigid and elastic results from finite plates using the present BEM with k₀=20 are compared against theoretical results for semi-infinite rigid [30] and elastic [1] plates using the Wiener–Hopf method. (Online version in colour.)

Figure 12 compares the scattered field owing to porosity alone against Wiener–Hopf results for a range of values of μ/k₀, where μ≡α_HK_R/R. For μ/k₀≪1, the impermeable limit is recovered [30,1], and the directivity of the scattered field is given by (5.3). If μ is comparable with k₀, porosity leads to reductions of the radiated sound relative to the impermeable limit. The reduction in radiated sound magnitude is accompanied by a progressive shift of the directivity shape from a cardioid to a dipole as μ/k₀ is increased, where directivity lobes from leading-edge modulation disappear in this low-frequency limit. Again, a direct comparison is difficult owing to the presence of the leading edge in the finite-plate problem; however, we note consistent magnitudes in scattered field strength between the semi-infinite results and finite-plate results for large Helmholtz number k₀ that support an indirect validation of the present computational framework.

Figure 12.

Edge noise directivity comparisons for impermeable and porous plates. See caption of figure 11. (Online version in colour.)

Data accessibility

All ΔPWL results presented in this work are accessible as electronic supplementary material of this article.

Authors' contributions

A.V.G.C. derived the mathematical model, wrote the code to obtain numerically the free-vibration modes, and generated all the numerical results. W.R.W. revised the mathematical model, wrote the BEM code for rigid impermeable plates and derived expressions to extend it to elastic plates with perforations. J.W.J. generated the semi-infinite plate results and contributed to the theoretical analysis. All authors worked on the development of the numerical code, on the writing of the manuscript, and on the analysis of results.

Competing interests

The authors have no competing interests.

Funding

The authors acknowledge the financial support received from Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, under grants no. 2013/03413-4 and 2014/05671-3, from Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under grant no. 470695/2013-7, and from the Science Without Borders program (project number A073/2013).